The challenges this causes are subtle. Arguments like those in [1]} seem to rely on a uniform constraint like \(\sup _{\theta _1 \in \Theta } \mathbb {P}(\hat{I}^+_{\tau } \ne I^*(\theta _1) \mid \theta =\theta _1) \leqslant \delta \) . Each choice of \(\theta _1\)  leads to a bound that is roughly of the form \(\mathbb {E}[\tau \mid \theta = \theta _0] \geqslant \Gamma _{\theta _0}^{-1}(M,\theta _1)\log (1/\delta )\) . Optimal results are derived by first maximizing over \(\theta _1\)  and then minimizing over \(M\) . This argument breaks in our case, because
(REF ) does not restrict \(\sup _{\theta _1 \in \Theta } \mathbb {E}[\Delta _{\tau } \mid \theta = \theta _1]\) , since the set \(\Theta \)  is infinite. We avoid this problem by showing Nature's worst-case behavior in Proposition REF  randomizes only over a finite number of states of nature. This is used at the very end of the proof.

